Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
158 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10624755 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10624737 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10624509 | C3 | 1 | 2 | yes | maximal, complemented |
| 4 | C32×3−1+2 | C32 | 1 | 1 | yes | characteristic |
| 5 | #6912 | — | 3 | 1 | no | — |
| 6 | #6910 | — | 3 | 1 | no | — |
| 7 | #6909 | — | 3 | 1 | no | — |
| 8 | C3×#732 | — | 3 | 3 | no | — |
| 9 | C3×AΣL(1,𝔽27) | — | 3 | 1 | no | — |
| 10 | C32⋊H(3,𝔽3) | — | 3 | 1 | no | — |
| 11 | #6866 | — | 3 | 2 | no | — |
| 12 | C34 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 13 | C3×3−1+2 | — | 3 | 1 | no | — |
| 14 | C3×3−1+2 | — | 3 | 1 | no | — |
| 15 | C3×3−1+2 | — | 3 | 1 | no | — |
| 16 | C3×3−1+2 | — | 3 | 2 | no | — |
| 17 | C3×3−1+2 | — | 9 | 3 | no | — |
| 18 | C3×H(3,𝔽3) | — | 3 | 3 | no | — |
| 19 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 20 | C3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 21 | C3×H(3,𝔽3) | — | 9 | 2 | no | — |
| 22 | C32×C9 | — | 3 | 1 | no | — |
| 23 | #733 | — | 9 | 3 | no | — |
| 24 | #732 | — | 9 | 9 | no | — |
| 25 | #731 | — | 9 | 3 | no | — |
| 26 | AΣL(1,𝔽27) | — | 9 | 3 | no | — |
| 27 | AΣL(1,𝔽27) | — | 9 | 1 | no | — |
| 28 | AΣL(1,𝔽27) | — | 9 | 2 | no | — |
| 29 | C32⋊C9 | — | 9 | 2 | no | — |
| 30 | C33 | — | 9 | 1 | no | — |
| 31 | C33 | — | 9 | 1 | no | — |
| 32 | C33 | — | 9 | 3 | no | — |
| 33 | C33 | — | 27 | 1 | no | — |
| 34 | C33 | AΣL(1,𝔽27) | 1 | 1 | yes | characteristic |
| 35 | C33 | AΣL(1,𝔽27) | 1 | 1 | yes | characteristic |
| 36 | C33 | #732 | 1 | 2 | yes | — |
| 37 | 3−1+2 | — | 3 | 3 | no | — |
| 38 | 3−1+2 | — | 9 | 2 | no | — |
| 39 | 3−1+2 | — | 9 | 6 | no | — |
| 40 | 3−1+2 | — | 9 | 1 | no | — |
| 41 | 3−1+2 | — | 9 | 1 | no | — |
| 42 | 3−1+2 | — | 9 | 1 | no | — |
| 43 | 3−1+2 | — | 9 | 1 | no | — |
| 44 | 3−1+2 | — | 9 | 1 | no | — |
| 45 | 3−1+2 | — | 9 | 1 | no | — |
| 46 | H(3,𝔽3) | — | 9 | 1 | no | — |
| 47 | H(3,𝔽3) | — | 9 | 1 | no | — |
| 48 | H(3,𝔽3) | — | 9 | 9 | no | — |
| 49 | H(3,𝔽3) | — | 9 | 1 | no | — |
| 50 | H(3,𝔽3) | — | 27 | 3 | no | — |
| 51 | H(3,𝔽3) | — | 27 | 6 | no | — |
| 52 | C3×C9 | — | 3 | 3 | no | — |
| 53 | C3×C9 | — | 9 | 2 | no | — |
| 54 | C3×C9 | — | 9 | 1 | no | — |
| 55 | C3×C9 | — | 9 | 1 | no | — |
| 56 | C3×C9 | — | 9 | 1 | no | — |
| 57 | C3×C9 | — | 27 | 2 | no | — |
| 58 | C32 | — | 3 | 1 | no | — |
| 59 | C32 | — | 3 | 1 | no | — |
| 60 | C32 | — | 3 | 2 | no | — |
| 61 | C32 | — | 9 | 1 | no | — |
| 62 | C32 | — | 9 | 2 | no | — |
| 63 | C32 | — | 9 | 1 | no | — |
| 64 | C32 | — | 27 | 3 | no | — |
| 65 | C32 | — | 27 | 6 | no | — |
| 66 | C32 | — | 27 | 3 | no | — |
| 67 | C32 | — | 27 | 3 | no | — |
| 68 | C32 | — | 27 | 1 | no | — |
| 69 | C32 | — | 81 | 1 | no | — |
| 70 | C32 | #6856 | 1 | 1 | yes | characteristic |
| 71 | C9 | — | 9 | 3 | no | — |
| 72 | C9 | — | 27 | 1 | no | — |
| 73 | C9 | — | 27 | 1 | no | — |
| 74 | C9 | — | 81 | 2 | no | — |
| 75 | C3 | — | 3 | 1 | no | — |
| 76 | C3 | — | 9 | 2 | no | — |
| 77 | C3 | — | 9 | 1 | no | — |
| 78 | C3 | — | 9 | 1 | no | — |
| 79 | C3 | — | 81 | 3 | no | — |
| 80 | C3 | — | 81 | 1 | no | — |
| 81 | C3 | #10624474 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #10624474 |
| commutator, Frattini, cosocle | C32×3−1+2 | C32 |
| LCS | G ▷ C32×3−1+2 ▷ C34 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C34 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C32×3−1+2 ▷ C34 ▷ C32 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624755 | C3 | semidirect |
| 2. #10624737 | C3 | semidirect |
| 3. #10624509 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624755 | 3 |
| #10624737 | 3 |
| #10624509 | 3 |